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We present high-order, finite element-based Second Moment Methods (SMMs) for solving radiation transport problems in two spatial dimensions. We leverage the close connection between the Variable Eddington Factor (VEF) method and SMM to convert existing discretizations of the VEF moment system to discretizations of the SMM moment system. The moment discretizations are coupled to a high-order Discontinuous Galerkin discretization of the Discrete Ordinates transport equations. We show that the resulting methods achieve high-order accuracy on high-order (curved) meshes, preserve the thick diffusion limit, and are effective on a challenging multi-material problem both in outer fixed-point iterations and in inner preconditioned iterative solver iterations for the discrete moment systems. We also present parallel scaling results and provide direct comparisons to the VEF algorithms the SMM algorithms were derived from.

We present a novel acceleration technique for improving the convergence of source iteration for discrete ordinates transport calculations. Our approach uses the idea of the dynamic mode decomposition (DMD) to estimate the slowly decaying modes from source iteration and remove them from the solution. The memory cost of our acceleration technique is that the scalar flux for a number of iterations must be stored; the computational cost is a singular value decomposition of a matrix comprised of those stored scalar fluxes.On 1-D slab geometry problems we observe an order of magnitude reduction in the number of transport sweeps required compared to source iteration and that the number of sweeps required is independent of the scattering ratio in the problem. These observations hold for an extremely heterogeneous problem and a 2-D problem. In 2-D we do observe that the effectiveness of the approach slowly degrades as the mesh is refined, but is still about one order of magnitude faster than source iteration., Comment: 4 pages